94年 - 94 國立中興大學_理學院科系(聯招)轉學考：微積分#112194

1.The upper sum = __(A)__. and the lower sum = __(B)__ if f(x) := |x|,x ∈ [-2,1] and P is the partition of [-2,1].

2.=__(C)__

3.=__(D)__

4.=__(E)__ for x＞ 0.

5.The radius of convergence of the power series =__(F)__

7.=__(H)__

(a) If f(x) :=x³ - 2x + 2, x ∈ , then f has a zero between -2 and 0.

(b) Consider two series. Suppose ≥ 0 and . If converges, then converges.

(c) If f : (-1,1) → is continuous at x = 0, then f is differentiable at x = 0.

(d) If f : (0,∞)→ (0,∞) is continuous and increasing on (0,∞)； then f(x)=∞.

(e) If f : (0,∞) →  is differentiable,i.e., f'(x) exists, , then f'(x) is continuous on .

(f) does not exist.

(g) The series  converges. 

(h) If converges, then converges. 

(i) If converges, then  converges.

(j) If f : [a,b] → is not continuous, then  f(x)dx does not exist.

(1) Evaluate .

(2) Find  at the point (-1,0) if 6x⁴+ 3x²y - 20xy² - 3y³ = 6.

(3) Find the gradient vector of the function f(x,y) := . Then find the equation of the tangent plane at (2,0).